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NPS ARCHIVE 
1960 
MCLAUGHLIN, D. 






i 









ANALOG SOLUTION OF CENTRAL! 
' ■ FORCE PROiUJfM 

DEAH H. MctAUG-HLIN 







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DUDLEY KNOX LIBRARY 

m«™S22 TGRADUATE SCHOOL 
MONTEREY, CA 93943-5101 



ANALOG SOLUTION OF 
CENTRAL FORCE PROBLEM 

by 

dean n. Mclaughlin 
lieutenant, united states navy 



/0>(?S Ale '£ 



ANALOG SOLUTION OF 
CENTRAL FORCE PROBLEM 

***** 



DUDLEY KNOX LIBRARY 
NAVAL POSTGRADUATE SCHOOL 
MONTEREY, CA 93943-5101 



Dean N. McLaughlin 



ABSTRACT 

Electronic analog computers have been used extensively for the 
solution and display of many dynamics problems. The majority of the 
problems worked with have been those involving linear differential 
equations with constant coefficients. In cases involving non- linear 
differential equations fewer solutions have been developed. This ha® 
been due mainly to the need of using non- linear elements in the com- 
puter circuits when setting up the solutions. 

One such problem, that of a mass moving under the action of a 
first power central force, is treated in some detail. The differen- 
tial equation is derived, the problem is scaled, and the circuitry 
developed. Solutions obtained by the use of the electronic analog 
computer are displayed and compared with solutions obtained by num- 
erical methods and errors and their sources are discussed. Finally 
there is an overall evaluation of the usefulness of analog computers 
to this sort of problem. In an appendix, a second practical dynamics 
problem is discussed, but a solution was not obtained due to lack of 
time available. 



TABLE OF CONTENTS 

Section Title Page 

1. Introduction - 1 

2„ Background 1 

3o The Problem and General Method of Solution 2 

4„ The Differential Equation of Motion 4 

5c Discussion of Equipment 5 

6o Computer Equation and Scaling 7 

7o Analog Computer Circuits 8 

8c Results 10 

9» Discussion of Discrepancies 23 

10c Conclusions 24 

Bibliography 26 

Appendix I 27 

Appendix II 37 

Appendix III 42 



iii 



LIST OF ILLUSTRATIONS 

Figure Page 

1. Force Diagram for Central Force Problem 4 

2„ Division Circuit 9 

3., Circuit Diagram 12 

4, Photograph of Computer Assembly 13 

5o Photograph of Problem Board 14 

• • • 

6o Recordings of, R, Z, R, and R 15 

7. Recordings of, 9 , R, R, and O 16 

8, Summary of Analog Results 17 
9o Radius vs Time Curves 18 
10. Angle vs Time Curves 19 
Ho Radius vs Angle of Rotation Curve 20 
12 Z vs Radius Curve 22 
A13. Table of Numerical Integration 30 
A14, Table of Calculated Values 32 
A15. Table of Calculated Values 33 
A16. Integration Curve, Part 1 34 
A17« Integration Curve, Part 2 34 
A18. Integration Curve, Part 3 35 



A19. & and r vs Time Curves 35 

A20. r vs Time Curve 36 

A21„ Table of Circuit Elements 41 



IV 



TABLE OF SYMBOLS AND ABBREVIATIONS 
(without subscripts) Constants in differential equation 
C, Capacitor (i = f, 1,2, 3, . . . . .) 
M Meg ohm 

R (without subscript) Voltage representing radius r 
R a Resistor (i = f ,1, 2,3, . . . . .) 

X A capital representing the voltage equivalant of a variable x 
?■ Output voltage of division circuit 

a Coefficient potentiometer value 
e Input voltage to an operational amplifier 

e Output voltage of an operational amplifier 

o 

f ^subscript) Element in feedback loop 

r Radius 

t Computer time 

t Problem time 
P 

oC^ Scaling factor (i » 1,2,3,...) 
& Angle of rotation 

UJ ^. Input voltage coefficient (i = 1,2,3,....) 

Capacitor 
Resistor 

Operational amplifier 
— ■ @ Coefficient potentiometer 




~^|^ f-» Function Multiplier 



1, Introduction 

This thesis presents the electronic analog solution to a non- 
linear dynamics problem which leads to the differential equation 

*" A fix I f 1 

An example problem is taken and the steps in reduc- 
ing it to a form suitable for an electronic analog computer^ hereafter 
referred to as an analog computer, are shown. The results are then 
compared with two solutions obtained by numerical methods. In Appen- 
dix III an equation of the form; >< =AX f 8X is 

discussed and the problems encountered in trying to obtain an analog 
computer solution are delineated. 

The writer wishes to express his appreciation for the assistance 
given him by Professor John E. Brock and to the Professors, particular- 
ly Professor 0. H» Polk, and the technicians of the Electrical Engineer- 
ing Department. The numerical solutions in Appendix I were contributed 
by Professor Brock. 
2. Background. 

Solutions for many dynamics problems have been established using 
analog computer and references can be found in the technical litera- 



ture. One such reference for a non- linear problem^, Analog Computer Solu- 

2 
tion of a Non- Linear Differential Equation, by H. G. Markey, (2) , was 

found but was only applicable in a general way to this investigation. 

It was considered that if a means could be found to display some 

of the classical problems encountered in early college dynamics on the 



i 

? d/dt is denoted by a dot placed above the variable operated on, 

"vers in parentheses refer to references in Bibliography., 



analog computer the following advantages would be obtained; 

(a) the general usefulness of the analog computer could be 
made more readily apparent; 

(b) in dealing with these problems attention could be focused 
on the dynamic principles leading to the governing differential equations 
and upon the mechanical significance of the results and not upon the mathe- 
matical difficulties in obtaining an analytical solution; 

Cc> in the case of those problems where analytical solutions 
have been obtained for certain particular parameter values and which 
therefore seem to be separated into many different cases the dynamical 
significance of which is not apparent, the general problem could be 
dealt with directly; 

(d) it would be possible to include normal dynamical in- 
fluences fsuch as energy loss due to pivot friction or windage) with- 
out so complicating the mathematics of the solution that the modified 
problem appears to be entirely different from the idealized problem,, 
In addition to the above it was desired to obtain these results 
using only the analog computers and their associated equipment, normal- 
ly available in an analog computer laboratory. 
3o The problem and general method cf solution. 

The problem considered was that of determining the subsequent- 
motion of a body weighing 1930 pounds, attached to a spring having a 
free length of five inches and a scale of ten pounds per inchj, when 
released with the following initial conditions,, At the initial instant 
the radius is four inches and its rate of change is zero; the polar 
anglej » ^ s zero and its rate of change is three radians per second. 



end of the spring is attached to a fixed point and the body 
■srmitted to slide without friction upon a horizontal plane. 
We will discuss the sequence of steps necessary for the solution 
a problem of this type 9 and then we will proceed with the solution. 
One might of course proceed with a full scale experimental program as 
a method of solution^ but eliminating this possibility we would s 

a. Using the principals of Mechanics arrive at one or more 

differential equations describing the motion. 
b„ Solve these equations 9 incorporating the starting condi- 
tions. This solution may be analytic^ numerical^ experi- 
mental (dealing with, possibly, scaled down mechanical 
variables) 9 or by means of an analog s in which one deals 
experimentally with variables of another type (such as 
electrical) which satisfy similar differential equations, 
Co Interpret the mathematical, experimental 8 or analog results 
in the proper mechanical light so as to arrive at a mean- 
ingful solution to the original problem. 
In this thesis 8 we are investigating the practicability of pro- 
ceeding immediately from the first step to a solution by use of stand- 
ard analog computer equipment. We do not have an analytical solution 
to the problem stated s although it is likely that one might be obtained 
in terms of elliptic functions and integrals. However^ in an appendix 
we will develop two different numerical solutions to the problem with 
which we can compare our analog solutions. 



The differential equation of motion. 
Figo 1 shows the body in a general position. The solid arrow re- 
presents the spring force F = 10(r-5), where r represents the radial 

s 

distance from the fixed point 0, The dotted arrows represent D'Alembert 

forces necessary for dynamic equilibrium. We see that 

F s +• Yn <X, 7 r: O 

Yn o*,q ~ O 
Now by kinematics, a^-r?-n& and <^-& --L sL. (n*& ) . From the 

second equation we see that ft & z C = Ccfi$£ t . This can also be seen 
from the fact that the angular momentum of the system about 0, namely 

Yn ft & , is invariable. From the first equations, we have 

10 (* -s) + F * 3 7j. (ji -jygfy-Q t and from this we iit-ih-lo+b-hd^z-O . 

Substituting & =. C /m^- we finally get ft = .£ Q.h +■ /O 

In our case s evaluating C at the initial instant we have C = 48, Thus 
we have as our set of differential equations: 

Now it is possible to perform some mathematical manipulations which 
simplify this system. In particular, it is easily possible to obtain a 
first integral of the first equation of the system. However 9 w© regard 
it as contrary to the spirit of this thesis, the purpose of which deals 
with the ready feasibility of making an analog computer solution of this 
system, to perform any such manipulations, and it is 




.<£? 



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Fig. 1 Force Diagram for Central Force Problem 

4 



this system with which we shall be directly concerned when we attempt 
the analog solution* The numerical solutions for this problem will be 
found in Appendix I. 
5* Discussion of equipment. 

Before taking up the solution of the problem, a description of 
the equipment used in the solution of this problem will be presented* 
It is assumed that the reader is already acquainted with the basic theory 
of the analog computer and with the usual circuitry used, such as summers, 
integrators, etc. The Handbook of Automation, Compution and Control, 
Vol, 2, E. M. Grabbe, (1), is a good reference for this information as 
well as for additional information on the equipment discussed below, 

A, Donner Analog Computer, Model 3000, 

This analog computer, which can be used for the quantita- 
tive solution of linear (and certain classes of non- linear) differential 
equations and transfer functions, contains ten DC operational amplifiers, 
any one of which can be used for addition, subtraction, multiplication or 
division by a constant, sign changing, or integration. Problems express- 
ed as differential equations are entered in terms of electrical components 
on a detachable problem board. Stability and accuracy are satisfactory 
for problem solution times up to 100 seconds or more which permits ac- 
curate recording with conventional pen recorders. (5) 

B. Donner Function Multiplier, Model 3730. 

This function multiplier consists of two multiplier 
channels and a regulated power supply. Each multiplier channel produces 
an output voltage which is accurately proportional to the instantaneous 
produce of two independent input voltages, where each input is either 



constant or varying with time Either input may be positive or negative 9 
so that four quadrant multiplication is provided,, The range of output 
and input voltages is plus and minus 100 volts; this being necessary to 
stay within the linear range of the operational amplifiers of the com- 
puter. To maintain the output voltage at 100 volts or less the Function 
Multiplier is designed to give an output voltage which is ,01 the product 
of the input voltages, f6) 

For the solution of the problem of this thesis two of these multi- 
pliers were used. They gave accurate results when used for straight 
multiplication although they do tend to drift over a period of time and 
have to be readjusted; this is a minor operation, however. When used in a 
division circuit, which is discussed in a later section^ the results obtain- 
ed were not as accurate^ however. It is believed, however, that this was a 
fault of the circuit and not of the ^unction multiplier bacause, as mention- 
ed above, the function multipliers gave quite accurate outputs when used 
for multiplication alone. 

C. Donner Function Generator, Model 3750. 

This variable base function generator is designed for use in 
conjunction with two external operational amplifiers to produce an output 
voltage which can be adjusted to approximate a desired single valued 
function of the input voltage. One operational amplifier is required for 
operation of the function generator and the other is used to accept the 
output signal at its summing junction. This amplifier may also be used 
for additional summing^ inverting, integrating or other operations. The 
function generator operates on the principal that the function can be 



approximated by a series of straight line segments, each segment being 
limited to a slope between plus and minus two volts per volt. The input 
and output voltages may vary between plus and minus 100 volts. (4) 

For the solution of the thesis problem it was desired to use 

3 
this function generator to generate the function 2304/r but it was found 

that the slope of curve for this function exceeded the capability of the 

equipment. This is duscussed further in Sec. 9. If it had been possible 

to use this function generator the two function multipliers would not have 

been required. 

6. Computer equation and scaling. 

To reduce our problem to a form suitable for the computer it is 
necessary to apply scaling factors. This was done using the methods out- 
lined in Basic Theory of the Electronic Analog Computer, by R. C. H. Wheel- 
er> (9). A brief summary of this process is presented here. 

The differential equation to be solved is first arranged so as to 

o* A. 

solve for the derivative of the highest order. In our case K- = H- 

n 3 

-BX-f-C . The equation is then scaled so that maximum value of each para- 
meter is represented by a voltage, close to but not exceeding 100 volts,, 
To do this scaling factors are assigned as shown by the following ex- 
ample: 



X = oc. 



X 



Here X is the computer voltage representing the variable x, and oC^ 
is its scaling factor. After being calculated the scaling factor is 
usually rounded off to facilitate computations. After suitable scaling 
factors are found, the equation is put into the form: 



«*R A -., - s*„R ,c 

.■'0' 






To develop the applicable circuits for the problem solution it is 
necessary to determine the values of resistance and capacity needed 
for each component of the circuit. Using the procedures in Wheeler' s 
book s (9), pages 2-10 we find, for example, that an operational amplifier 
when used as a summer has an output voltage ^ -~L U - J ,G / -f-w^C.^ -4 ) 

or in our case Rs-(wi SL _ ^K^i^H . If we now let 6^ - c*-^ Kf - 

where a. is a coefficient potentiometer setting and R_ and R. are re- 

x r f i 

sistances, we can establish the relationship C*--^ _ UJj, '\ ,_ 

It should be noted here that an R with a subscript, R„. refers to a resis- 

tor and . R. without the subscript refers to the voltage representing the 

variable r, the radius of the problem. Now the above relationship can 

be solved for a . For integrators the relationship is Co^ - ^Z^=- » 

where C f refers to a capacitor. 

7. Analog computer circuits. 

In Appendix II the calculations for scaling the differential equa- 
tions of our problem are presented. After scaling we have the follow- 

ing equations: ft = __, _ ^ g ■> ^ 0Q Q 

K 3 

Before solving this problem on the analog computer two main decisions 

3 2 
have to be made: first how to calculate R and R , and second how 

3 * 2 

to develop the terms Z = 576,000/R and & = 30,000/R . It was 

hoped at first that the terms for Z and could be developed using func- 
tion generators but as mentioned previously this proved unsatisfactory,, 
Thus it was expedient to use the division circuit shown in Fig. 2„ 

8 



H 



«< 

<- 



■ 



h 



\AAA- 




Division Circuit (6) 
Fig. 2 



With this circuit Z = 100R| X 



The factor 100 results from the 

2 3 

output of the multiplier being .01 Y£. If we now let Y = R and X 

= constant, using the above relationship we should be able to develop 

Z = 576,000/R 3 . 

We know from the parameters of the problem that when r is 4 S 

R should be 20 volts. If we then put this value through two function 

2 3 
multipliers we come up with K R . As this value is small, .8 volt, we 

multiply it by a factor of two using an operational amplifier and then 

put it into the function multiplier of the division circuit. Also using 

this value of the voltage for R we can calculate the value Z should 

have, in this case 72 volts. With these values we can now solve for a 

value of X so that with an input of 1.6 volts for Y and the calculated 

value of X , Z will be 72 volts. Solving for X: 



X - 



- / !\ 









> 



/oo R t 



Now letting R ~ equal 10M and R equal 1M, we find that an X of 
11.5 should be used. (It was found that resistances of 10M and 1M worked 
better than resistors of 0.1M and 1M) . This same procedure was applied 
to & and the corresoonding voltage, X. was found to be 30 volts. 



It should be noted here that another method for determining Q 
presents itself, that of multiplying Z by R/19.2. By doing this the 
second division circuit could be eliminated and only another multiplica- 
tion 3 with its more accurate results, required. This method was tried 
and it was found that for some unexplained reason Q passed through 
zero and became slightly negative. As a result of this S oscillated 
instead of increasing smoothly from zero to a maximum value. For this 
reason the division circuit for developing Q was used. 

After the above determinations were made, the circuit of Fig. 3 was 
assembled and computations made. In assembling the circuit the values of 
the a's calculated in Appendix II were adjusted for the actual values of 
resistances and capacitors used, e.g., 1.005 M actual resistance vs. 
nominal value of 1M. Figs. 4 and 5 are photographs of the setup used 
and shows the relative simplicity of the final setup for solution of the 
problem. 
8. Results. 

After assembling the circuit of Fig. 3, it was found that to obtain 
the desired values of voltage for Z and the values of the input vol- 
tages calculated for X and X- had to be adjusted. For X s a value of 
20 volts, and for X a value of 33 volts was required. Once these ad- 
justments were made s the computing runs were made and the results are 
shown on the Brush Recordings of Fig. 6 and Fig. 7, These recordings were 
all made using a paper speed of 5/mm/sec and with varying voltage scales as 
shown on each trace. These results are also summarized in the table of Fig. 
8. From these results curves were plotted and then compared with the re- 
sults obtained by the numerical solutions^ as shown in Figs. 9 S 1Q» & il„ 

10 



In analysing the results each term will be considered separate- 
ly„ Considering r first it is seen that the maximum value of 38 ob- 
tained agrees with the maximum value of the numerical solution but that 
the minimum value of - 12 c 5 is lower than the - 15.26 of the numerical 
solution. This latter discrepancy is attributed to the actual values 
obtained for Z and will be discussed later. 



11 



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Summary of Analog Results 


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26 


2 


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81 


9o7 


23 


o92 






9o5 


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3 


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23 


- 7 


- 3o5 


30 


9c6 


20 


o 80 


kk 


8o3 


L0 


68o7 




o7 


22 


-12 


- 6 


75 


9o0 


18 


c72 






11 


75.5 


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21 


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- 7o5 


67.5 


8.1 


15 


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53 


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-17 


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19 


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Fig. 8 

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30 



20 



10 



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Solution #2 



1+6 8 

KadiuSj inches 



ii+ 



Fig 11, Radius vs Angle of Rotation 



20 



Fo-r r the maximum value obtained was 9,7 and the minimum 
zero when r was a maximum and a minimum. This agrees well wi 
the numerical solution where the maximum value was 9,6, Considering r 
w® see from Fig, 9 that the analog values are slightly higher at all vain 
than the r°s of the numerical solution. This error is not considered 
excessive. 

The largest discrepancies appear when we consider ©„ As can be 
seen in Figs„ 8 9 10 s and 11 the analog value reached its maximum for 
the first apse (point of greatest distance from the center of attraction' 
rapidly and then remained constant for a period of time. Here as with 
r the discrepancies are considered to be caused by the values obtained 
for ©„ 

Considering the problem overall^ the more significant results ob- 
tained appear to agree rather well with the values obtained by the 
numerical solutions. The major discrepancies appear when the part of 
the circuit handling the division is considered. As can be seen from 
FigSo 6 and 12 for Z, and Fig, 7 for © 9 the outputs of these division 
circuits change rapidly to a small negative voltage and then remain re- 
latively constant for a period of time. We can also see from Fig, 10 
that the division circuit does not do what theoretically it should,, Thus 
for either parameter the minimum voltage desired s when r is a maximv?'.- 
is never obtained. With Z 9 this term is small when compared witli 
others in summing for r and the effect is not pronounced. With ( 
however this defect has a more pronounced effect and % is not dev« 
in the smooth curve desired. 



21 



9, Discussion of discrepancies. 

The discrepancies found in the above problem solution were attri 
ed to the division circuits used. No satisfactory answer could be found 
as to why the desired divisions could not be obtained „ It is known 
that a circuit such as this develops a certain amount of noise. That is,, 
the function multiplier has a certain amount of noise inherent in its 
output and that if this is put through an operational amplifier this 
noise is amplified. The Handbook of Automations, Computation and Control,, 
Vol, 2 S by Grabbe (1) discusses this briefly and mentiors that a small 
capacitor placed in parallel with the multiplier will help to alleviate 
this problem. This was tried but did not give satisfactory results. 

As mentioned previously, if a function generator could have been 
used 8 the circuitry could have been simplified, i,e„ 9 no function multip- 
liers would have been required. With the Donner function generator the 

3 

slope of the function 576 8 000/ R s for low values of R s exceed the maxi- 
mum of two volts per volt permitted by the device. One other type of 
function generator was tried. This was an Autograft XY plotter converted 
to a function generator by replacing the recording pen with a' pick-up 
coil and plotting the desired function with a conducting ink„ However 8 
with this arrangement the desired range of voltages could not be obtain- 
ed. 

Still another type of function generator that might have proved 
satisfactory s if it had been available, is the photo- former type. This 
type of function generator operates as follows. The basic piece of 
ment is a cathode -ray tube. An input voltage is applied between 



23 



.. 1 deflection plates of a cathode ray tube th* stable 
amplifier. The voltage between the vertical deflect i< is pla 

as the output voltage. This voltage is made to vat i funct . 
of the input voltage by a feed-back arrangement which forces -. lectrom 
beam to follow the boundary of an opaque mask placed over the lower por- 
tion of the cathode-ray screen. Thus as the spot on the eathode-ray tube 
screen emerges from behind the mask a photocell in front of the tube ap- 
plies an error voltage across the input terminals of the vertical de£! i - 
tion d-c amplifier s so phased that the beam is forced downward toward the 
face of the mask. Therefore if the mask is shaped to represent the functis 
being generated the spot will follow this curve and deliver an output 
voltage proportional to the input voltage. This type of function genera- 
tor is said to be very accurate in developing many functions. (3) 
10. Conclusions. 

Considering the results obtained from this problem (keeping in mind 
that indeed it is but a single problem) 9 it was found that a "typical" non- 
linear dynamics problem can be set up on an analog computer. However this 
type of set-up is not done rapidly or easily. Considerable thought has 
to be given as to what type of equipment shall be used and what kind of 

-cults are necessary. Because they require the use of various types 
of non- linear computer accessories the circuits become very sensitive and 
results accurate to the degree normally expected from the analog computer 
may not be obtained. Care has to be taken in selecting scaling faetors s 
where powers and roots are involved,, to avoid over- loading the operation- 
al amplifiers. It was found, however 8 that the function multipliers used 



24 



square and cube R gave quite accurate results^ even at It- 
they were kept balanced. 

In setting up a problem of this type it will usually be found 

3 

that there will be one key term to be developed s such as the A/R of 

this problem,. Once a way is found to develop or represent this term 
the remaining computer setup is routine and with patience and luck a 
solution can be obtained. 



2y 



PHY 

1, -. M. Grabbe s S„ Ram© and D. E. Wooldridge s Handbook of Auto- 
mat ion, Computation and Control^, Volume 2 9 Computers and Data 
Processings John Wiley & Sons,, 1959. 

2, H. Mo Paynter 9 A Palimpsest -~ fr he Electronic Analog Art s Geo., A. 

Philbrick Researches,, Inc„j, 230 Congress St. 9 Boston,. Mass^ 1955. 

3, Go A Kom and T. M„ Korn g Electronic Analog Computers 9 2nd Edition^ 
McGraw-Hill Book Co., 1956. 

4» Operating Handbook Donne r Model 3750 Variable Base Function Genera- 
tor 8 Donner Scientific Co. 

5, Operating Handbook Donner Model 3000 Analog Computer, 

6. Operating Handbook Donner Model 3730 Function Multiplier, 

7o J, L. Synge and B. A, Griffith, Principals of Mechanics » 2nd Edition,, 
McGraw-Hill Book Co. 8 1947, 

8o N. C. Riggs 9 Applied Mechanics, The MacMillan Co., 1930, 

9o R Co H. Wheeler, Basic Theory of the Electronic Analog Computer,, 
Donner Scientific Co., 2829 - 7th St . , Berkeley,, Calif., 1955. 



26 



APPENDIX I 

Numerical Solutions for Central Force Problem 

The statement of the problem is given in Sec 3 on page 2„ Re- 
presenting this information in mathematical terms s we have F 

spring 

2 

lQCr-5) lbs. and m = 1930/386 = 5 lbs sec /in. s and initially (at 

time = 0) we have r = 4 inches s r = Oj, = S and 0-3 
radians / sec o Since energy is conserved 8 we have 

where E>, T s and V are expressed as energy per unit mass in units 
of it. */ sec 2 . Here we have used V- L. '/*-«-( 

Using the initial conditions to evaluate E 9 we have 



u 



13 



i 

Ac apses n Apsidal radii are 

given by n -$v - Substituting 

and rationalizing w@ gets 

7 y -/Oft 3 -j-nS'/j 1 -- • 3 ~ 



/ 3 - yr/7' 4 -f-zJb 
one root is 4 we obtain (/?-</) ( 
This can have only one positive root„ Synthetic division indicates a 
root of approximately 13.2 and using Newton's method; 



2 7 



<x, ^ /3 - £6l) -_ )3 _ (-") _^ /3,/j- 

To find the apsidal angle and r and as functions of time,, we 
resort to a numerical procedure since the integral involved is not 
elementary. Returning to fundamentals we have: 

,/7 — ftp* - ~a^-*~3 

; o /,* ^ 03£T _^ y-/0 
We also know r 1 = 4 and r„ = 13.144, Now using an iterated 

« 

Integration, a curve of r = r(t) is assumed such that r = at 
the end points (apses') „ The apsidal time "<Z is divided into n equal 
intervals /r\ ; C being as yet unknown. We will use n = 6 S although 
a larger n will give a more accurate result „ 

CO „ 

Assumed values of v* are selected for each epoch. Values of r 
are calculated and integrated with the condition/?-^ at £T *■ ° 
This should yield/7 -0 at t-^ 8 but there is an error €L „ We remove 
this error by using a correction curve which is essentially A ft- (2.^ -i-l ^j^- 
expressed however in appropriate form for and obtained by numerical 

m 

integration. This arises from assuming that the error in r is due 
to an error in r which must be essentially parabolic in nature 9 vanish- 

28 



Ing at the end points since the apsidal distances are known . The rest 
of the calculations are self explanatory and lead to the curved 
shown in Fig, 9» 



29 



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I9fiq QNZI 



Second Numerical Method 
From the expression for E, /f^ & +"A~ +-%(p~ v ^ 

Upon substituting & /A , we get f^ as a function of A 2 
and thus can construct a curve of 77 as a function of Jr\ (We take 
the positive branch of the square root so as to deal with the period 
during which r is increasing from 4 to 13.145 inches.). Also we have 
J~1 ~ £3oV _ ^ /j -f~ J so that we can construct a curve of yy 

as a function of r, and this relation shows that fi - O when r is 

& a. 

approximately equal to 7,62. Having curves of both /? and f) as 

functions of r, we can construct a curve of h as a function of A 

The differential of time may be written in either of two w«ys 

c JLu. c^> <$JX t and this permits us to write 

ft rr 

Aid) ^ 7 A Oj./vst) 

^ r 4^ ^ r ^ ^ r ^i 

so as to avoid infinite values for the integrands. These calcula- 
tions can be carried out by numerical methods as shown on the follow- 
ing pages and & is found to be 1.4863 seconds s which agrees with the 
value of calculated in the first numerical solution. 



31 







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36 



APPENDIX II 
Scaling Equations for Central Force Problem 
The basic equations for this problem are: 

a -- *%*- 

The initial conditions at time t = are: 

m~--/ J w - o J a-- o J 3 --* j 

and we also know from the numerical solution that the approximate 
maximum values of each of the parameters are: 

Also from the numerical solution we know that the time from apse to apse 

is about 1.46 seconds. For our solution we will select 0~-2>Tf » (1.5 

revolutions) as we are only interested in the initial aspect. Knowing 

the maximum values and using the relationship x - oC^ X s as explained 

in Sec. 6, we can now solve for the scaling factors. 

<• - /3./V ,/3/V /, U^<X--,X Ky^-G^V 



/0O 



h -Si—— — 



p/.- - _3_ ^ ,03 / , LcL (*-£ - /C y &y^ j 0*V 

^ " /oo 
For time scaling we wish to slow the problem time down so we assume 
~C . - S'^p s giving a period for the computer of 7.3 seconds,, apse to 
apse. 

37 






I 






! 






Now using the above scaling factors we can proceed to scale 
basic equations for the computer: 

(x A 3 «») 



.'> 







/•^ TIP 







The equations are now in the form f% - —(JaJ. S — dJ., ^ ^~CU,/^°J 
re Z = 576, 000/ R , and ^ = 30 S 000/R . Because Z and & are 
developed by the division circuits and 20 is a constant voltage their a* 
values (coefficient potentiometer settings) are each 1. Thus their 
corresponding resistors are all 1M. To determine the value of a- we 
use the relationship c <^j; :_ °^LJ&- . Equating 6*4, to c> ^-£ 2 -~ / ^ 
and letting R„ and R _ equal 1M S we find a to be .8. 



38 



To obtain R and R we must integrate ^j ** cx-Ls> and 

We therefore scale these equations as follows: 



'C 






,tt'y 









, 2 ' w 



Now using the relationship ^<' \ - — ^~ we solve for a,. 

as shown below. The resulting values are .833 and .48 respectively 

using the resistors and capacitors shown. 

To complete our scaling we must now determine tie a value 
for C7 . cr is found by integrating ( &cx<=p . Scaling this 



equation we find 



1 fit, ^ - SilA Ja 



*& 



^---^1 CjJt 






39 






o/^ - 









The problem is now ready for the computer. The table on the 
following page summarizes the values for all resistors and capacitors 
used for this problem. 



40 



1 


'able of Circuit Elements 


Amplifier 


Circuit Element 


Remarks 


(See Fig. 3) 


^and Value 




1 


R7I = 1M 
R72 - 10M 


Forms -Z 




R 1 "= 1M 






r 2 : im 






r 3 : im 




*,'- 


R f2 : im 

a^ -1.0 
a 2 s 0.8 
83 = 1.0 


* * 

Sums + R 




Rl ■ = IM 




3 


C f ^ = ly-f 
a^ - 0.833 


- f +R dtp = -R 




R^ = 2M 




k 


ac; = O0I4.8 


- f -R. dtp = +R 




Rp = 5M 




5 


C f5 = 1/jf 

a g a 0.33 


-j-*a V 


7 


R81 = IM 
R32 = 10M 


* 

Forms — @ 




R 6 = 0.1M 




9 


R - 0.2M 
f9 

a£ =1.0 


2K 2 R-* 



Fig. A21 
41 



APPENDIX III 
The Spinning Top 
During this investigation, some attention was also directed to 
the problem of the spinning top; that is a symmetrical top with one 
pc int fixed under the action of gravity. Using the notation of Synge 
a 



nd Griffith, (7), the equation * ^ % O^^fr - **7*P0 '> l ) -^'A^ 
was reduced to the form of }< ' #X 3 ~ C X Z -h c * +" "^ 
A sample problem was then selected from Applied Mechanics by N, C 
Riggs (8), which consisted of a gyroscope being released with & equal to 
60 degrees and the subsequent motion being an oscillation between 60 and 
82 degrees while precessing at a variable rate,, 

With this problem, as with the central force problem, the powers 
of x could be obtained using function multipliers. The key to the 
solution however consists of developing the square root of X . This 
should be possible using either a function generator or a division cir- 
cuit similar to that of the central force problem. Using the function 
generator was tried first. A curve of Y = 10 X l/2 9 X and Y 
being arbitrary voltages from zero to 100,, was calculated and set into 
the function generator. It was then found on scaling the problem that 

to prevent the term CX from exceeding 100 volts it was necessary to 

•2 
assign to X a maximum voltage of approximately nine volts. Then 



•n the circuit was assembled it was found that this voltage was too 
small for the function generator to operate. 

When the use of the function generator proved to be unsatisfactory 
the use of a division circuit was attempted. To use this it was assumed 



42 



that the following relationship was valid., We know that for this 

circuit Z - J_0C R^ X. Now letting Y equal Z and then so 

R 2 Y 2 \/Sl 

for X we find that X = R ^ Z , or Z equals 10 X " , with 

100^ 

R - R = 1M. This arrangement was then put into the circuit and 

computations attempted. It was found that the operational amplifier 
generating X went from its initial value through zero and then over- 
loaded, i.e., the voltage X (representing x = Co<, cr ) s did not 
vary as it should since the circuit was unstable. 

After the above attempts failed it was decided that this particular 
problem could not be scaled satisfactorily for an acceptable analog 
computer solution within the time available. 



43 



\